2004 Mazda RX8 Engine/Idling Problems, Help?

The bad shaking makes me think engine mounts, but that does not explain your other problems. You may need a new alternator, get it tested at autozone/checkers/etc. The coolant tank is known to get a faulty level sensor. Go get your CEL read at autozone while they are checking the alternator and battery and see what code pops up. Post the code here or do a search on rx8club.com.